Normalized defining polynomial
\( x^{22} - 9 x^{21} - 84 x^{20} + 897 x^{19} + 2755 x^{18} - 38122 x^{17} - 41461 x^{16} + 903508 x^{15} + 173110 x^{14} - 13126552 x^{13} + 3288569 x^{12} + 121271266 x^{11} - 53972002 x^{10} - 717482927 x^{9} + 340654907 x^{8} + 2680513897 x^{7} - 1010896265 x^{6} - 6090097243 x^{5} + 1071244813 x^{4} + 7718191075 x^{3} + 746605202 x^{2} - 4228140005 x - 1647782009 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(305334364114002390216524630254855801940178013=23^{20}\cdot 37^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $105.20$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $23, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(851=23\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{851}(1,·)$, $\chi_{851}(519,·)$, $\chi_{851}(73,·)$, $\chi_{851}(75,·)$, $\chi_{851}(334,·)$, $\chi_{851}(591,·)$, $\chi_{851}(593,·)$, $\chi_{851}(147,·)$, $\chi_{851}(223,·)$, $\chi_{851}(739,·)$, $\chi_{851}(36,·)$, $\chi_{851}(554,·)$, $\chi_{851}(556,·)$, $\chi_{851}(813,·)$, $\chi_{851}(110,·)$, $\chi_{851}(369,·)$, $\chi_{851}(371,·)$, $\chi_{851}(630,·)$, $\chi_{851}(186,·)$, $\chi_{851}(443,·)$, $\chi_{851}(445,·)$, $\chi_{851}(702,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{895021} a^{20} + \frac{359815}{895021} a^{19} + \frac{437909}{895021} a^{18} - \frac{338984}{895021} a^{17} + \frac{390010}{895021} a^{16} - \frac{225831}{895021} a^{15} + \frac{405685}{895021} a^{14} + \frac{248547}{895021} a^{13} + \frac{398386}{895021} a^{12} - \frac{154468}{895021} a^{11} - \frac{396000}{895021} a^{10} - \frac{273860}{895021} a^{9} - \frac{421950}{895021} a^{8} + \frac{399}{6533} a^{7} + \frac{257095}{895021} a^{6} + \frac{179255}{895021} a^{5} + \frac{66922}{895021} a^{4} - \frac{403889}{895021} a^{3} + \frac{60172}{895021} a^{2} - \frac{19908}{895021} a - \frac{197168}{895021}$, $\frac{1}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{21} + \frac{78294159738476390521056006444703851818165937717207496800057347853646709}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{20} - \frac{10445007469734589114595420601497842044019841823195811405769598101196403460217}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{19} - \frac{61392061061880083059262320417244670461559944211873546757990202323830597560004}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{18} + \frac{62157252692870921744047820974263414998615062150377386629229438662269723599485}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{17} - \frac{66675866352570737988999176869715252837053233687027867959843734740581602324485}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{16} + \frac{37004159841942883859530747319304833074767700816404080687798595509248283197812}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{15} + \frac{62588284552844981124188642159582501850154839937386298578976367944052846739480}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{14} - \frac{53361684200750738099103503020972249100076302963206296035941562980703263574100}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{13} - \frac{25593325533159369177361521563490961929311190022533975633302597379105511978337}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{12} - \frac{20687949888992200234765174425375715817923880710702658819496941204459260747835}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{11} - \frac{21325744127507906944358027573616598178858220039058961338845522149376257467119}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{10} - \frac{38610196417169377777107269569819620050285361475666246004183598723216761721387}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{9} - \frac{79915807685102573313198235777719624901474288214942154414129345021741325741521}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{8} + \frac{18204385399790828627094364311967071304445524672075877948078086438710073599479}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{7} + \frac{56174079953792666348737305470004878088522949033173649410021052855600756447354}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{6} - \frac{37116510684180674926141153983784808703339124785069929919824986847039341053857}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{5} + \frac{27547834211870048775082066559208978494806597960855006922911885978820606939666}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{4} + \frac{12783817687008815186612632188739352563650753923163950296443340597865149209113}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{3} - \frac{26744438727015134492673821303647450877970916152351450226466288279097295218737}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a^{2} - \frac{72738348477293212713894193409023903774959854608222506754649798313959528818786}{160449753851598888434142434134739346661211981160197543028874658977479820801301} a + \frac{21149764508150057516375274067740359591517213019415457749138170100531440749139}{160449753851598888434142434134739346661211981160197543028874658977479820801301}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $21$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1888088815265438.8 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{37}) \), \(\Q(\zeta_{23})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | R | $22$ | $22$ | R | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 23 | Data not computed | ||||||
| 37 | Data not computed | ||||||